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Differential form: Section of a bundle of antisymmetric tensors. Geodesic: General
geometrical equivalent of a straight line in Euclidean geometry. Metric tensor:
Fundamental tensor field for prescribing metrical geometry on a manifold.
B. Metric Tensors A metric tensor on an n-manifold M is an element g e T\M, that
is, a section of T\M g: M-* T°2M :x\^>gx € TXM* ® TXM* where gx: 7xil/ x TXM -» R
: («, u) l-» &,(«, u) satisfies the conditions 1. Symmetry: gx(u, v) = gx(v, u). 2.
There is a dual metric tensor g* 6 Y/jA/ with g*: TXM* x 7"XA/* -» R : (a, /3)H- g* (a
, 0) with coordinate expression £,* = (** 3, ® djh and (#«) = (*„)-' as matrices, well
defined since det g^ 4 0. 3. There is an isomorphism Y?M =« Y^M of tangent ...
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Liquid Alkali Metals 1 John F Schenck
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Liquid Crystal Devices 41 Manifold Geometry
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